Ken's POTW

Bob's House, and Productive Numbers Revisited

Bob lives on a long street and has noticed that the sum of the house
numbers up to his own house, but excluding it, equals the sum
of the numbers of his house to the end of the road. If the houses are
numbered consecutively, starting from 1, what is the number of Bob's
house? Assume there are less than 1000 houses on the road.

Find all numbers i, such that i is
equal to the sum of the cubes of its digits.